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Europhysics Letters (EPL)

Quantumness of ensemble via coherence of Gram matrix

Yuan Sun, Shunlong Luo, Xiangyun Lei

Published May 1, 2021• 7 citations
Publisher

Abstract

The Gram matrix of a set of vectors, which encapsulates the relations between the constituent vectors, plays an important role in the exploration of both geometric and information-theoretic aspects of quantum state space. In view of its usefulness and importance, we study the Gram matrix of an ensemble of pure states (a set of pure states with a prior probability distribution) and reveal its fundamental properties. We highlight and exploit the fact that the Gram matrix of an ensemble can be formally regarded as a bona fide quantum state. The key idea is to employ coherence (with respect to the computational basis) of the Gram matrix (regarded as a quantum state) to quantify quantumness of the corresponding ensemble. In particular, we propose to use the l 1 -norm of coherence and the relative entropy of coherence, of the Gram matrix, as two significant quantifiers of quantumness. We illustrate the effectiveness and power of these quantifiers of quantumness by evaluating them for several important ensembles arising from quantum measurement and quantum cryptography. We further compare the quantumness based on the Gram matrix with various other quantifiers of quantumness in the literature, and show that although they are closely related in general, they have subtle differences and capture different aspects of ensembles, which shed light on the complexity of quantum ensembles.

DOI
10.1209/0295-5075/134/30003

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The Gram matrix of a set of vectors, which encapsulates the relations between the constituent vectors, plays an important role in the exploration of both geometric and information-theoretic aspects of quantum state space. In view of its usefulness and importance, we study the Gram matrix of an ensemble of pure states (a set of pure states with a prior probability distribution) and reveal its fundamental properties. We highlight and exploit the fact that the Gram matrix of an ensemble can be formally regarded as a bona fide quantum state. The key idea is to employ coherence (with respect to the computational basis) of the Gram matrix (regarded as a quantum state) to quantify quantumness of the corresponding ensemble. In particular, we propose to use the l 1 -norm of coherence and the relative entropy of coherence, of the Gram matrix, as two significant quantifiers of quantumness. We illustrate the effectiveness and power of these quantifiers of quantumness by evaluating them for several important ensembles arising from quantum measurement and quantum cryptography. We further compare the quantumness based on the Gram matrix with various other quantifiers of quantumness in the literature, and show that although they are closely related in general, they have subtle differences and capture different aspects of ensembles, which shed light on the complexity of quantum ensembles.
Abstract The Gram matrix of an ensemble of pure states can be regarded as a quantum state, and the quantumness of the ensemble can be quantified by the coherence of the Gram matrix. By using the affinity between mixed states, the concept of Gram matrix of pure-state ensembles can be extended to the one of mixed-state ensembles. By utilizing the generalized α - z -relative Rényi entropy of coherence of Gram matrices, we present a new quantifier of quantumness of mixed-sate ensembles and further reveal its peculiar properties. To illustrate our quantumness of mixed-sate ensembles, we also calculate the quantumness for some detailed mixed-sate ensembles by deriving their analytical formulae.
Abstract A quantum measurement (e.g., positive-operator valued measure, or more generally, instrument) performed on a system state yields a post-measurement ensemble (a family of outcome states together with the corresponding outcome probabilities), which may have quantumness due to the impossibility to perfectly discriminate the constituent states. In order to characterize information contents and structure of a measurement, it is desirable to investigate the post-measurement ensemble. In this work, we study post-measurement quantumness (i.e., quantumness of the post-measurement ensemble) by introducing a Gram matrix summarizing the interaction between a measurement and a state, and explore its implications and consequences. Gram matrix of an ensemble can be regarded as a quantum state (density operator) in an auxiliary system with a computational basis labelling the measurement outcomes. Our basic idea is to exploit coherence of the Gram matrix (with respect to the computational basis of the auxiliary system) and employ it to quantify post-measurement quantumness . We propose several quantifiers of quantumness and reveal their basic properties. We further evaluate explicitly quantumness of measurements arising from several important quantum channels, and illustrate a comparison between them. This provides a method for investigating measurements from the perspective of coherence of Gram matrix.
The resource theory of coherence quantifies the quantum properties of states and their role in quantum information processing. Extending coherence theory to ensembles, the natural objects in quantum communication, remains an active research direction, with their operational potential still being explored. In this work, we establish a maximally coherent ensemble in every finite dimension: the continuous ensemble of pure states distributed according to the normalized Haar measure attains the tight upper bound on basis-free relative entropy ensemble coherence. We then prove that ensemble coherence bounds the effective informational advantage of postselective measurements over the Holevo quantity for two equiprobable pure states. Finally, for the qubit B92 QKD protocol, we show that the coherence of Alice's ensemble upper-bounds the secret key rate in the asymptotic regime. These results give ensemble coherence a quantitative operational meaning in quantum communication and cryptography
Given an orthonormal basis BBB in a ddd-dimensional Hilbert space and a unital quantum operation E\mathcal{E}E acting on it, one can define a nonlinear mapping that associates with E\mathcal{E}E a $d\ifmmode\times\else\texttimes\fi{}d$ real-valued matrix that we call the coherence matrix of E\mathcal{E}E with respect to BBB. This is the Gram matrix of the coherent part of the basis projections of BBB under E\mathcal{E}E. We show that one can use this coherence matrix to define vast families of measures of the coherence-generating power (CGP) of the operation. These measures have a natural geometrical interpretation as separation of E\mathcal{E}E from the set of incoherent unital operations. The probabilistic approach to CGP discussed in P. Zanardi et al. [Phys. Rev. A 95, 052306 (2017).] can be reformulated and generalized, introducing, alongside the coherence matrix, another $d\ifmmode\times\else\texttimes\fi{}d$ real-valued matrix, the simplex correlation matrix. This matrix describes the relevant statistical correlations in the input ensemble of incoherent states. Contracting these two matrices, one finds CGP measures associated with the process of preparing the given incoherent ensemble and processing it with the chosen unital operation. Finally, in the unitary case, we discuss how these concepts can be made compatible with an underlying tensor product structure by defining families of CGP measures that are additive.
Building on the geometric formulation of quantum mechanics, we develop a coherence theory for ensembles that exploits the probability-phase structure of the quantum state space. Standard coherence measures quantify superposition within density matrices but cannot distinguish ensembles that produce the same mixed state through different distributions of pure states. First, we introduce the probability-phase mutual information I(P;Φ)I(P;\Phi)I(P;Φ), which measures statistical correlations between measurement-accessible probabilities and measurement-inaccessible phases across an ensemble. Then, we prove this satisfies the axioms of a coherence monotone, establishing it as a bona-fide measure of ensemble-level coherence. Eventually, through the definition of the \emph{coherence surplus} δC≥0\delta_{\mathcal{C}} \geq 0δC​≥0, we show how ensemble coherence relates to, but exceeds, density-matrix coherence, thus quantifying structure lost in statistical averaging.